The HVZ theorem for a pseudo-relativistic operator
The author studies the localization of the essential spectrum for the pseudo-relativistic Brown-Ravenhall operator (B-R Op.). Let \(H=\sum_{k=1,2}(D^{(k)}_0+ V^{(k)})\), where \(D^{(k)}_0:= \alpha^{(k)}p_k+\beta^{(k)}m\) (with \(p=-i\nabla_x)\), and \(V^{(k)}:= -\gamma/|x_k|\) \((\gamma= Ze^2)\). Let \(V^{(1, 2)}:= e^2/|x_1- x_2|\). The B-R Op. is defined by \(H^{BR}= \Lambda_{+,2}(H+ V^{(1,2)})\Lambda_{+,2}\), where \(\Lambda_{+,2}= \Lambda^{(1)}_+\Lambda^{(2)}_+, \Lambda^{(k)}_+\): projector onto the positive spectral subspace of \(D^{(k)}_+\). The B-R Op. \(H^{BR}\) has a block-diagonal form with one nonvanishing entry, \(h^{BR}\), defined by means of \((\phi_+, H^{BR}\phi_+)= (u, h^{BR} u)\). He proves the HVZ Theorem: Let \(h^{BR}= \sum_{k=1,2}(T^{(k)}+ b_{1m}^{(k)})+ V^{(1,2)}\), \((T= T^{(1)}+ T^{(2)}\,T^{(k)}:= E_{p_k})\), be the two-electron B-R Op. with potential strength \(\gamma<\gamma_{BR}= 2/(\pi/2+ 2/\pi)\) (\(-\gamma\): coefficient in \(b_{1m}^{(k)}\)), and let \(h^{BR}= T+\alpha_j+ r_j\), \(j= 0,1,2\), be its two-cluster decomposition. Then the essential spectrum of \(h^{BR}\) is given by \(\sigma_{\text{ess}}(h^{BR})= [\Sigma_0,\infty)\). Here \(\Sigma_0= \min_j(\text{inf\,}\sigma(T+\alpha_j))= \min\{\text{inf\,}\sigma(T+ b_{1m}^{(1)})\), \(\text{inf\,}\sigma(T+ v^{(1,2)}\}\).
scientific article; zbMATH DE number 5151295